Is it possible to construct a sequence like this?

[kindlychung]

Homework Statement

Is it possible to construct a sequence like this? If yes, please construct one, if not, give a proof. See the attached pic for requirements for the sequence to be constructed.

Homework Equations

NO

The Attempt at a Solution

I know how to construct a sequence that has subsequences that converge to a finite number of limits, such as:
1, -1, 1, -1, ...
1, 2, 3, 1, 2, 3, ...

This problem brings the sine function to my mind, but since R is uncountable, it's not possible to build a sequence that contains terms that has a 1-1 correspondence with sin(x).

I might as well ask: what is the cardinality of $$N \times N$$?
$$card N \times N = card N ?$$
In that case I could just repeat the sequence {1, 1/2, 1/3, ...} over and over again.

 

[╔(σ_σ)╝]

I don't know what you are referring to when you talk about the sine function considering the sequence in question only takes of rational numbers; which are countable.

Anyway, NxN has the same cardinality has N.

 

[Office_Shredder]

In that case I could just repeat the sequence {1, 1/2, 1/3, ...} over and over again.

You can't just stick it after itself over and over again

{1,1/2,1/3,...,1,1/2,1/3,...,1,1/2,1/3,..}

isn't a sequence. Which natural number does the second 1 correspond to? You essentially need to find a bijection between N and NxN in order to write down the sequence you want

 

[kindlychung]

If NxN has the same cardinality has N, then by definition there is a bijection between N and NxN.

 

[╔(σ_σ)╝]

If NxN has the same cardinality has N, then by definition there is a bijection between N and NxN.

There are many of them.

In fact, a one to one function on N would do.